Volume 18, pp. 81-90, 2004.

Abstract

The polynomial numerical hull of degree $k$ for a square matrix $A$
is a set in the complex plane designed to give useful information
about the norms of functions of the matrix; it is defined as
\[
\{ z \in {\bf C}:~\| p(A) \| \geq | p(z) |~\mbox{ for all polynomials $p$
of degree $k$ or less} \} .
\]
In a previous paper [V. Faber, A. Greenbaum, and D. Marshall,
The polynomial numerical hulls of Jordan blocks and related
matrices, Linear Algebra Appl., 374 (2003), pp. 231–246]
analytic expressions
were derived for the polynomial numerical hulls of Jordan blocks.
In this paper, we explore some consequences of these results.
We derive lower bounds on the norms of functions of Jordan blocks
and triangular Toeplitz matrices that approach equalities as the
matrix size approaches infinity. We demonstrate that even for
moderate size matrices these bounds give fairly good estimates
of the behavior of matrix powers, the matrix exponential, and the
resolvent norm. We give new estimates of the convergence rate
of the GMRES algorithm applied to a Jordan block. We also derive
a new estimate for the field of values of a general Toeplitz matrix.